NAND With Equal Arguments

Theorem

Let $\uparrow$ signify the NAND operation.

Then, for any proposition $p$:

$p \uparrow p \dashv \vdash \neg p$

That is, the NAND of a proposition with itself corresponds to the negation operation.

Proof by Tableau

Proceed by the Tableau method:

$p \uparrow p \vdash \neg p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p \uparrow p$	$\mathrm P$	(None)
2		1	$\neg \left({p \land p}\right)$	By definition	1
3		1	$\neg p$	$\mathrm {Idemp}$	2

$\neg p \vdash p \uparrow p$

Line		Pool	Formula	Rule	Depends upon
1		1	$\neg p$	$\mathrm P$	(None)
2		1	$\neg \left({p \land p}\right)$	$\mathrm {Idemp}$	1
3		1	$p \uparrow p$	By definition	2

$\blacksquare$

Proof by Truth Table

We apply the Method of Truth Tables:

$\begin{array}{|ccc||cc|} \hline p & \uparrow & p & \neg & p \\ \hline F & T & F & T & F \\ T & F & T & F & T \\ \hline \end{array}$

As can be seen by inspection, the truth values under the main connectives match for all models.

$\blacksquare$